The Existence of Quasimeromorphic Mappings in Dimension 3

Abstract

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{3}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (i.e torsion) elements are uniformly bounded. This is accomplished by developing a technique for meshing distinct fat triangulations while preserving fatness. We further show how to adapt the proof to higher dimensions.